Spline finite strip analysis of the buckling and vibration of composite prismatic plate structures

A spline finite strip capability is described for predicting the buckling stresses and natural frequencies of vibration of prismatic plate structures which may be of composite laminated construction with arbitrary lay-ups. The plate structures may have general boundary conditions. The capability embraces analyses based on the use of first-order shear deformation plate theory and of classical plate theory, and utilizes substructuring procedures which include the use of superstrips. The theoretical development is not detailed since the present paper reports a very direct extension of a theoretical study developed for the analysis of single plates in an earlier paper in this Journal. A considerable range of buckling and vibration applications is documented and comparison of spline finite strip numerical values of buckling stresses and frequencies is made with results generated using the semi-analytical finite strip method and, in some cases, the finite element method. Buckled and vibrational mode shapes are presented for some applications.     

An advanced analysis of axisymmetric plastic sheet bending including transverse shear deformation

Finite deformation theory for axisymmetric elastic-plastic sheet bending is developed. The theory incorporates transverse shear deformation by adopting the extended Kirchhoff-Love's hypothesis, i.e. during bending deformation, a line element normal to the undeformed mid-surface is allowed to change the angle between itself and the mid-surface while its straightness is retained. A new idea, called “equivalent curvature”, is proposed which plays a similar role to the curvature in conventional plate bending theory but incorporates the effects of transverse shear deformation. Numerical calculations based on this theory have been performed for two examples of sheet-forming processes, i.e. the hydrostatic bulging of a circular sheet and the U-type sheet-bending process. Results show that the proposed theory can predict more precise information concerning the forming processes for a moderately thick sheet than the conventional sheet bending theory which ignores transverse shear deformation.     

Stochastic analysis of compositionally graded plates with system randomness under static loading

This paper presents an investigation of the stochastic bending response of moderately thick, compositionally graded plates with uncertainties of low variability and subjected to lateral load and uniform temperature change. System parameters such as the thermal and mechanical material properties of each constituent material, volume fraction index, and load intensity are taken as independent random variables. The basic formulations are based on Reddy's higher-order shear deformation plate theory and a semi-analytical method. A first-order perturbation technique is employed to obtain the second-order response statistics-mean and variance of the flexural deflection of plates with various boundary conditions. Typical results are presented for two types of plates containing functionally graded materials made of metallic phase Ni and ceramic phase Al₂O₃. It is found that the response sensitivity of the plate is very much dependent on the material composition. Variations in Young's modulus and lateral load have dominant effects on the stochastic characteristics compared to other random parameters. The deflection dispersion of compositionally graded plates shows the so-called “non-intermediate” characteristic even when thermal loading is absent.     

Modeling mechanical response of intumescent mat material at room temperature

Intumescent mat material is widely used to support ceramic substrates in catalytic converters and behaves very much like hyper-foam material under compressive loading. Experiments show that compressive loading curves depend on the ram speed and the number of cycles. The unloading curves show different slopes and paths that depend less on the ram speed and number of cycles. The slopes of the unloading curves decrease as the plastic strain increases; this is referred to as “softening” in this study. The effects of rate, softening, and plastic deformation must be considered to model the mechanical response of intumescent mat material. Finite deformation theory is applied with a multiplicative decomposition of the deformation gradient tensor. The developed theory is implemented as an implicit finite element algorithm in ABAQUS^TM/STANDARD. The necessary material parameters are extracted from experiments. Numerical simulations show good agreement with experiments.     

Closed-form solution for wedge cutting force through thin metal sheets

A simple kinematic model is developed which describes the main features of the process of the cutting of a plate by a rigid wedge. It is assumed in this model that the plate material curls up into two inclined cylinders as the wedge advances into the plate. This results in membrane stretching up to fracture of the material near the wedge tip, while the “flaps” in the wake of the cut undergo cylindrical bending. Self-consistent, single-term formulas for the indentation force and the energy absorption are arrived at by relating the “far-field” and “near-tip” deformation events through a single geometric parameter, the instantaneous rolling radius. Further analysis of this solution reveals a weak dependence on the wedge angle and a strong dependence on friction coefficient. The final equation for the approximate cutting force over a range of wedge semiangles 10° ≤ θ ≤ 30° and friction coefficients 0.1 ≤ μ ≤ 0.4 is: F = 3.28σ₀(δ_t)^0.2l^0.4t^1.6μ^0.4, which is identical in form and characteristics to the empirical results recently reported by Lu and Calladine [Int. J. Mech. Sci.32, 295–313 (1990)].This analysis is believed to resolve a controversy recently developed in the literature over the interpretation of plate cutting experiments.     

Stochastic perturbation-based finite element for deflection statistics of soft core sandwich plate with random material properties

The effect of random variations in material properties of laminated sandwich plates on the transverse deflections is investigated. An improved higher-order plate model is proposed earlier by the authors, which satisfies the transverse shear stress continuity conditions at the layer interfaces including the zero transverse shear stress conditions at the plate top and bottom surfaces. The theory assumes the variation of in-plane displacements to be cubic with discontinuities in the transverse shear strains at the layer interfaces, while the transverse displacements varies quadratically across the core thickness, thereby including transverse normal deformation of the soft core. The core is considered to behave as a 3-D elastic medium. To obtain the second-order statistics of deflections of sandwich plate, a stochastic C⁰ finite element (FE) based on the first-order perturbation technique is developed, where the lamina properties are considered as basic random variables while the other system properties are assumed to be deterministic. The performance of the improved stochastic laminated sandwich model is demonstrated through comparison of mean and standard deviations (SDs) of deflections obtained through independent Monte Carlo simulations and by comparison with results available in literature.     

EFFECT OF SOME MODIFICATIONS TO OXLEY'S MACHINING THEORY AND THE APPLICABILITY OF DIFFERENT MATERIAL MODELS

Oxley's machining theory has recently been extended[1] to accept material property inputs in the form of widely used constitutive models such as the Johnson-Cook and MTS material models. In the process, additional modifications have been made to the model to improve its self-consistency. For instance, the shear force is obtained from the total work of deformation, thereby eliminating the unknown parameter η, and the hydrostatic pressure at the tool-chip interface is calculated considering the gradient in temperature in addition to the gradient in strain. This study is aimed at understanding the effect of these modifications separate from the changes due to the introduction of the new material models by comparing results obtained using Oxley's original model to that obtained with the above modifications. We also compare results obtained using different constitutive models for AISI 1045 to the experimental results of the “Assessment of Machining Models” effort.     

Application of a comprehensive dynamic cutting force model to orthogonal wave-generating processes

The dynamics of a cutting process are very complex in nature. They involve not only the changes of plastic state in the intensive shear zone of the chip formation process but also the elastic behaviour of work material surrounding the plastic deformation zone, especially in the vicinity of the tool nose region. As an extension to the previous developments in formulating the shear angle oscillation in dynamic cutting (D. W. Wu, Development of dynamic shear angle model for wave-generating processes based on work-hardening slip-line field theory. Int. J. Mech. Sci. 29, 407–424, 1987; D. W. Wu, Governing equations of the shear angle oscillation in dynamic orthogonal cutting. Trans. ASME J. of Engng for Indust. 108, 280, 1986), a comprehensive dynamic cutting force model has been developed from the mechanics of the cutting process by taking into account the equilibrium of forces in the primary and secondary plastic deformation zones and the redistribution of the contact stress inside the workpiece in the vicinity of the tool nose region.The model has been tested through a computer simulation for orthogonal wave-generating processes. By reference to existing experimental evidence, the theoretical predictions show generally good agreement with the test results.     

Elastic small deflection analysis of annular sector Mindlin plates

An analytical method is developed for the bending response of annular sector Mindlin plates with two radial edges simply supported, and exact solutions are presented in the form of Levy-type series. Several different boundary conditions on the two circular edges are considered, viz. simply supported-simply supported, clamped-clamped and free-free. Numerical results for the case of uniform loading are presented to indicate the effect of shear deformation on the deflections and stress resultants at various points in the plate. Twisting stress couple and transverse shear stress resultant distributions along and near the edges of the plate are illustrated graphically, and the principal differences between the results predicted by Mindlin's plate theory and classical thin plate theory are discussed in detail. Results obtained with the present exact analysis may serve as references for approximate solutions and, especially, as a ‘shear locking’ test for thick plate finite element analysis.     

On the analysis of sheet metal wrinkling

An analysis for the prediction of wrinkling in curved sheets during metal forming is presented. Using a local approach, similar to that employed for conventional forming limit diagram representations, we construct “wrinkling limit curves” (WLCs) which represent the combinations of the critical principal stresses for wrinkling in curved sheet elements. Wrinkling limit curves are first determined using a bifurcation analysis for plastic buckling in short-wavelength shallow modes. A study of the effects of material properties and sheet geometry on the critical conditions for wrinkling is carried out. We then analyse the effects of geometric imperfections on wrinkling. This analysis is based on the implementation of a finite element scheme. The influence of nonproportional loading is also investigated. In our analysis the material is assumed to be isotropic, elastic-plastic with the plastic part modelled using both J₂ deformation theory and J₂ flow theory of plasticity.     

Towards a new theory of cold rolling thin foil

The theoretical framework is developed for a new theory of cold rolling thin metallic foil. Unlike previous theories, the work rolls are allowed to deform to a non-circular profile and finite regions of no-slip between strip and work rolls are allowed to occur in the roll bite. The theory predicts that plastic reduction occurs near entry and near exit of the roll bite, separated by a central region where the strip does not suffer reduction and does not slip relative to the work rolls. As the reduction is decreased to zero the theory reduces to essentially the Johnson and Bentall theory [J. Mech. Phys. Solids 17, 253 (1969)] for the onset of plastic reduction in a strip. At large strip thicknesses and finite reductions the new theory approximates the Bland and Ford theory [Proc. Inst. Mech. Engrs 159, 144 (1948)] of cold rolling.     

Free vibration of cantilevered symmetrically laminated thick trapezoidal plates

To account for the effect of transverse shear deformation, the p-Ritz method incorporating Reddy’s third-order shear deformation theory has been developed for the vibration analysis of cantilevered, thick, laminated, trapezoidal plates. In the p-Ritz method, a set of uniquely defined polynomial functions, consisting of the product of a two-dimensional function and a basic function, are used as the admissible trial displacement and rotation functions in the Ritz minimization procedure. The energy integral is formulated based on Reddy’s third-order shear deformation theory. From the p-Ritz method, the governing eigenvalue equation is derived which is used to compute the vibration frequency parameters and mode shapes of the laminated plate. Convergence and comparison studies have been presented to demonstrate and verify the accuracy of the results.     

A finite element formulation based on an enhanced first order shear deformation theory for composite and sandwich structures

A finite element formulation based on an enhanced first order shear deformation theory is developed to accurately and efficiently predict the behavior of laminated composite and sandwich structures. An enhanced first order shear deformation theory is systematically derived by minimizing the least-squared energy error between the first order shear deformable plate theory and a higher order shear deformable plate theory. In this way, the strain energy of a higher order theory is transformed to that of the Reissner-Mindlin plate theory. This minimization procedure yields a relationship between them that is also used to improve the accuracy of predicted stresses and displacements. The key feature of the proposed theory is in that it can be implemented to commercial FEM packages by simply changing the input, and the results obtained can be also enhanced by post-processing them via a differential quadrature method. Thus, a proposed finite element formulation can be widely used in various application problems. Through numerical examples, the accuracy and robustness of the present formulation are demonstrated.     

Theoretical and experimental results of the plastic and strain-hardening behaviour of En8 at a controlled rate

This investigation contains two theoretical analyses of the plastic behaviour and work-hardening characteristics of medium carbon steel (En8). The first of these analyses employs Perzyna's visco-plastic constitutive law for strain-rate sensitive and work-hardening material behaviour, and the second uses the strain-rate independent theory of Prandtl and Reuss. Both theories gave good agreement with the time histories of the axial stress, the plastic strains and the stress trajectory. However, these theories gave an under-estimate of the shear stress decay versus time. It is also evident from the present study that neither of these theories could account for the lack of coaxiality between the plastic strain-rate and deviatoric stress vectors observed in the experimental results for the present bilinear deformation path.The objective of the present study was to test the ability of these two models to predict the overall response of a real material to biaxial loading at a controlled rate under a deformation path with an abruptly changing direction. A preliminary comparison was made of the strain-hardening response according to the two models for a material without Lüders strain region. Then existing experimental results of a bilinear deformation path of quasi-static twisting at a reference rate of 10⁻⁶ s⁻¹ beyond the initial yield in torsion (to γ = 10γ₀ where the Lüders strain was exhausted), followed by extension at a constant strain-rate with γ held constant, performed earlier by Meguid and Malvern [1], were used for comparison with these models. It is hoped that this information will guide the designer and the user of stress analyses programs towards more realistic material input data.     

Isogeometric bending analysis of composite plates based on a higher-order shear deformation theory

This research paper presents an isogeometric plate finite element formulation for analysis of thick composite plates. Isogeometric finite element method which is based on non-uniform rational B-splines (NURBS) basis functions, is a novel numerical procedure developed to bridge the gap between CAD and FEM modeling of structures. In order to investigate the behavior of isogeometric plate elements under static loading, plate kinematics is based on third order shear deformation theory (TSDT) of Reddy, which is free from transverse shear locking. This paper discusses accurate transverse stress recovery procedures for TSDT isogeometric finite elements. Numerical experiments with quadratic, cubic and quartic elements are presented and obtained results are compared to other available ones.     

Numerical study of the flange earring of deep-drawing sheets with stronger anisotropy

A Barlat–Lian anisotropy yield function is introduced into a quasi-flow corner theory of elastic–plastic finite deformation and the elastic–plastic large deformation finite element formulation based on the principle of virtual velocity and the discrete Kirchhoff triangle plate shell element model. The focus of the present researches is on the numerical simulation of the flange earring of deep-drawing process of circular sheets with stronger anisotropy, based on which, the schemes for controlling the flange earring are proposed.     

Different degree of reduction and sliding phenomenon study for three-dimensional hot rolling with sandwich flat strip

Symmetric rolling of 3D sandwich flat strips with thermal-elastic–plastic coupled model was studied under the assumption of an elastic roller and the condideration of heat transfer. Aluminum–copper sandwich flat strips were used in this study.The numerical model of symmetric rolling for 3D sandwich flat strip with thermal-elastic–plastic coupled model was developed based on the large deformation–large strain theory, the update Lagrangian formulation and the incremental principle. Besides, flow stress was considered as the function of strain, strain rate and temperature. The theoretical model of finite element method containing the two-order strain rate formulation acted as the basis for determining the convergence of simulation results.The contact surface between the aluminum and copper for the sandwich flat strip was also discussed. First of all, the contact face between the aluminum and copper was assumed that it would be fixed without sliding. Symmetric hot rolling of the aluminum and copper sandwich flat strip was analyzed. A slide criterion was then introduced to study the shear stress states of the contact face between aluminum and copper of sandwich strip, which was used to compare the relation between the maximum shear stress and the yielding shear stress on the contact face. If the maximum shear stress of aluminum or copper is smaller than the yielding stress of aluminum or copper respectively, sliding does not occur on the contact face. On the contrary, the sliding may occur on the contact face between aluminum and copper.Three different degrees of reduction were simulated in this study to analyze the states of shear stress on the upper aluminum strip and lower copper strip close to the contact face. Finally, it finds that the sliding on the contact face between aluminum and copper may occur around certain degree of reduction. The average rolling force of the simulation result was compared with experimental data [8] to verify the simulation results.     

An exact solution for buckling of functionally graded circular plates based on higher order shear deformation plate theory under uniform radial compression

In this research, mechanical buckling of circular plates composed of functionally graded materials (FGMs) is considered. Equilibrium and stability equations of a FGM circular plate under uniform radial compression are derived, based on the higher order shear deformation plate theory (HSDT). Assuming that the material properties vary as a power form of the thickness coordinate variable z and using the variational method, the system of fundamental partial differential equations are established. A buckling analysis of a functionally graded circular plate (FGCP) under uniform radial compression is carried out and the results are given in closed-form solutions. The results are compared with the buckling loads of plates obtained for FGCP based on the first order shear deformation plate theory (FSDT) and classical plate theory (CPT) given in the literature. The study concludes that HSDT accurately predicts the behavior of FGCP, whereas the FSDT and CPT overestimates buckling loads.     

A new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate

A new hyperbolic shear deformation theory taking into account transverse shear deformation effects is presented for the buckling and free vibration analysis of thick functionally graded sandwich plates. Unlike any other theory, the theory presented gives rise to only four governing equations. Number of unknown functions involved is only four, as against five in case of simple shear deformation theories of Mindlin and Reissner (first shear deformation theory). The plate properties are assumed to be varied through the thickness following a simple power law distribution in terms of volume fraction of material constituents. The theory presented is variationally consistent, does not require shear correction factor, and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions. Equations of motion are derived from Hamilton's principle. The closed-form solutions of functionally graded sandwich plates are obtained using the Navier solution. The results obtained for plate with various thickness ratios using the theory are not only substantially more accurate than those obtained using the classical plate theory, but are almost comparable to those obtained using higher order theories with more number of unknown functions.     

Thermal buckling behavior of laminated composite plates: a finite-element study

In this paper, the thermal buckling behavior of composite laminated plates under a uniform temperature distribution is studied. A finite element of four nodes and 32 degrees of freedom (DOF), previously developed for the bending and mechanical buckling of laminated composite plates, is extended to investigate the thermal buckling behavior of laminated composite plates. Based upon the classical plate theory, the present finite element is a combination of a linear isoparametric membrane element and a high precision rectangular Hermitian element. The numerical implementation of the present finite element allowed the comparison of the numerical obtained results with results obtained from the literature: 1) with element of the same order, 2) the first order shear deformation theory, 3) the high order shear deformation theory and 4) the three-dimensional solution. It was found that the obtained results were very close to the reference results and the proposed element offers a good convergence speed. Furthermore, a parametrical study was also conducted to investigate the effect of the anisotropy of composite materials on the critical buckling temperature of laminated plates. The study showed that: 1) the critical buckling temperature generally decreases with the increasing of the modulus ratio E _L/E _T and thermal expansion ratio α _T/α _L, and 2) the boundary conditions and the orientation angles significantly affect the critical buckling temperature of laminated plates.